(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "The theory of the construction of ordnance with special reference to the resistance of guns to tangential strain"

^ ^>^ :> 3l> -Bl> >i^^ ^^^- 



X)^LX)D>3> .3»S> ^3>^^4>, 















i>^ 






^3::>j^x^ 






:>»"-?i^ 



!»">■:> 



§!fc>.:n. 









7^ ?>:t^^^- 






r>jj^3t:i 



^>3^.1>5^-^- 



:0i>ai: 



>:5>^j 













8 



/ 



y 



L 



THE THEORY 



OF THE 



CONSTRUCTION OF ORDNANCE 



WITH SPECIAL REFERENCE TO THE 



RESISTANCE OF GUNS 



TO 



TANGENTIAL STRAIN. 



PEEPAEED FOE USE AT THE U. S. NATAL ACADEMY. 



ANNAPOLIS, MD, 

1883. 



TTASHINaTON: 

GOYERNMENT PEINTINa OFFICE. 

1883. 



5318 1 



INTRODUCTION. 



The following pages consist principally of a translation of extracts 
from Yirgile's work on the "Eesistance des tubes metalliques, simples 
ou composes, avec application a la construction des benches a feu." 

The deduction of the equations in Part I, which are used in the ap- 
plication of the theory in Part II, has greatly simplified the discussion 
of resistance to tangential strain. This part of the work has been done 
by Lieut. C. A. Stone, and is used instead of Yirgile's method. 

Some few changes and omissions have been made in the original, as 
the object in view was simply to provide a suitable text for purposes of 
instruction. 

H. B. EOBESOK, 
Commander JJ. S, H. 
3 



THE THEORY OF THE CONSTRUCTION OF ORDNANCE. 



FIRST PART. 

THEORY. 

CHAPTER I.— GEIS^EEAL PEIKCIPLES. 

1. Let a right rectangular prism be submitted to a tension parallel to 
one of its systems of edges so tliat the strain shall be uniformly spread 
over the transverse section in the proportion of iJ units of force for a 
unit of surface. The edges on which the force acts will be lengthened, 
while those that are perpendicular to the direction of the force will be 
shortened. 

While the force p does not exceed a certain limit, the changes of form 
which it determines will not be of a permanent character ) they will 
disappear when the force ceases to act. It is then said that these 
changes of form are elastic. 

If the force j? exceeds this limit, the prism, after the force has ceased 
to act, will not exactly resume its former dimensions. The changes of 
form thus produced are said to be permanent. Permanent changes of 
form, not apparent at first, increase more rapidly than the changes due 
to elasticity. A moment even occurs when the latter diminish and dis- 
appear almost entirely at the instant of rupture. 

If the force _p is a pressure instead of a tension, the contrary effects 
will be observed, that is to say, the edges parallel to^ will be shortened 
while the others will be elongated. Otherwise the prism will pass 
through the same changes of form, at first elastic, then i^ermanently 
changed and elastic at the same time, at last almost entirely i^erma- 
nently changed, and this series of phenomena will be terminated by the 
crushing of the prism. 

2. The value of j? for which these changes of form cease to be entirely 
elastic is called the limit of elasticity. There is for each substance and 
for each particular state of this substance two limits of elasticity, one 
referring to tension and the other to pressure. 

When ]) does not exceed the limit of elasticity, the elongation or 
compression determined by this force is proportional to it, whether 
applied in its own direction or at right angles 5 but beyond this limit 
the change of form increases more rapidly than p. In theory it is not 



considered that the pressures or tensions exceed the limit of elasticity. 
This reservation admitted, let i be the elongation or compression pro- 
duced by the force p acting in the direction of the elongation or com- 
pression 5 1^ is called the modulus of elasticity; this ratio is ordinarily 
1 

expressed by E. 

The modulus of elasticity is the force, theoretically, which must be 
applied to a unit of transverse section to double the length of the prism. 

3. Wertheim, by experimenting with diaphanous bodies, found that 
the modulus of elasticity of extension was the same as for compression. 
We take for granted that it really is so. At the same time it is to be 
remarked that the experiments of physicists give smaller values of the 
modulus of elasticity, in the case of certain metals, for compression than 
for extension. These differences may be caused by the manner of action 
of the forces which govern the constitution of the bodies experimented 
upon. 

4. Wertheim found by a series of carefully conducted experiments 
that an actual limit of elasticity did not exist ; that is to say, that all 
strains, no matter how feeble, provided that they are of long duration, 
cause a permanent elongation. On the other hand many authors believe, 
and it is generally conceded, that the changes of form increase more 
rapidly than the forces which cause them, even when they are far from 
the limit where a permanent change of form is easily distinguished; so 
that there is, strictly speaking, no actual modulus of elasticity. 

However, it is certain, as we have already seen, that permanent 
changes of form are at first extremely small in comparison with the 
changes that are elastic; at least this is true for all bodies which may 
be termed in any degree elastic. It is equally true that the curve of 
the change of form as a function of the force which produces the change 
is, to a greater or less extent, identical with the tangent at the origin 
of co-ordinates. It therefore results that we may consider the theory 
based upon the existence of a limit and modulus of elasticity, as exact 
for small tensions and pressures, but as susceptible of furnishing only 
approximate results for greater values. The strain or pressure to which 
this theory can be applied without too great an error will be the prac- 
tical limit of elasticity , and the mean modulus corresponding to the press- 
ures and tensions comprised in this limit will be taken as the modulus 
of elasticity. 

Let E = the interior radius of the tube ; 
E'= the exterior radius of the tube ; 
P = the distance of any point from the axis ; 
t = the hoop-tension at any point ; 
p = the radial pressure at any point ; 
T and T', P and P^ the values of t and pforp = B> and p = IV. Then 
P and P' will be the internal and external pressures. 








7ksm*0 



9 ^^ » 



-- .T^'^-TK. 




Consider the section aa'^ through the axis of length 
I. The section of the metal on either side may be gene- 
rated by the line I moving parallel to the axis, with 
the rate dp from />=R to p:=W. Therefore the re- 
sistance of the tube to rupture along the given section 
will be 



2i£ tap 



The bursting effort will be 2 Z (PE — Y'W) ; therefore 

/ ^^^ = PE~P^R' . 



(1) 



Let t=f'(p)) substituting this value of t in equation (1), and 
integrating, we have 

/ (EO — / (E) = P E — P^ E'. 
Since E and E' are independent of each other, we can write 

f(W) = -V'n' 

f{p) = -PP' 
Differentiating this equation, with reference to /?, gives 



or in general 



/'(,)=-j,-,|=* 



(2) 



The solution is effected by means of two partial solutions. First: 

d 1) 
Let— ^=m=a constant in (2), then 7^=0, and equation (2) becomes 

t——p=m. Substituting this value in (1) we have 



m / ''^/>=PE-P^E'=m(E'-E), 



which satisfies the equation. 

For the second solution, let t=p in equation (2). Then that equation 
becomes 

P dp 






Let 
then 



2 dp 



dp_^2a 

'dTp^y 



a 



(3) 
(4) 



8 
Substituting this value of t and^? in equation (1), 



•/■ 



dp _ r 1 1 1 



Integrating, we have 

1 r 1 111 1 

c-2 LR^-^ E''^-^ J~ E^-2'" E'--2* 
To satisfy this equation we make c=3; which gives in equation (4) 

a 

The value of c may also be determined by substituting the value of 
a from (4), a =jt?/>'^^ in equation (3), and integrating j 



therefore 






log|=2 1og^, 



whence 

Therefore p varies inversely as the square of p. Adding the two 
partial values of *, we have 



and also for^, 



t = ~-^m (5) 



P = -,-m (6) 

From equations (5) and (6) we have, 

f-jp = 2m, * + ^ = 7? ..... (7) 

Therefore 

(^ + i>)^' = (T+P)R' (8) 

and ^_^=:T-P (9) 

From equations (7) the following properties are deduced : 
the difference between the hoop-tension and the radial pressure is 
the same at each point ) and the sum at any point of the hoop-tension 
and the radial pressure varies inversely as the square of the distance 
of the point from the axis. 






^'CU, [aXTiaxV. 



To determine a, substitute the value of c in equation (3), and integrate, 
and we have 






Let i? = P^ and ^ = E^ ; then 



a = 



(P - PQ E^^ B^ 
W -E^ 



(10) 



Substituting this value of a in the preceeding equation, solving for 
^, and reducing, we have 

PE^(E^^-^^) + P^E^^(^^-E^) ... 

To determine m : From (6) 

substituting the value of a from (10), and reducing, 

PE2-P^E'2 



. W^-H^ 



(12) 



PAET II.— CHAPTER I. 

APPLICATION OF THE THEORY. 

Tubes and guns consisting of one piece of metal. 

§ 1 . The equations deduced in Part I, although they suppose the 
elasticity of a body to remain constant in all directions, will answer for 
most cases that occur in practice. It is, however, true that the elas- 
ticity of the same body varies both as regards direction and position, 
though these variations are generally inconsiderable when applied to 
metals. 

2. — Resistance of homogeneous tubes. 

Assuming equations (8) and (9) of Part I, 

(^M t-p =T-P 

in which T and P represent the tension and the pressure at the surface 
of the bore, t and p the tension and pressure of any point situated at 
a distance p from the axis, E the radius of the bore. If T' and P' repre- 
sent the tension and the pressure at the exterior surface, and E^ the 
radius of this surface, we shall have, putting p = E^, 



w{ 



(T' + P0E^2^(T + P)E2 

T'_P/ =:T-P. 



Combining (a) and (b), we derive: 

i{t+p)p^={T + ^^)n- 

^ M t-p = T^-P^ 

Since P' only stands for the pressure of one atmosphere, it may be 
neglected. Putting, then, P^ = 0, eliminating T, T^, snidp in the four 
equations {a) and (c), and solving with reference to t, we have: 

E'2 
^ ~ ^ E^2 

We see by this formula that the maximum value of t corresponds to the 
minimum value of p', that is to say, of ^ = E. It is then at the surface 
of the bore that the metal undergoes the greatest tangential strain. It 
results from this that we will assure the resistance of the tube (to 

10 



11 

racture) by determining the pressure P so that the tension T, at the 
surface of the bore, shall not exceed the tensile strength of the metal. 

3.—Calculatio}i of resistance. 

Putting ^ = R, the preceding formula beconfes 

R'2 ^ 

-R^ + 1 
T = P^^,-- 

R^ ~ ^ 
from which we have 

(1) P=T ^^ ^ 



R^24.R2 



This result may be obtained directly by making P' = 0, in equations 
(&), by eliminating T' and solving with reference to P. 

Formula (1) was given for the first time by Lame ; it readily permits 
the calculation of the pressure P, which corresponds to a given tension 
of metal. If T is the limit of tension which is not to be exceeded, P will 
represent the resistance of the tube, these two quantities being referred 
to the unit of surface. 

If in formula (1) we make E' vary from the minimum value of R togl^ 
or else R from R' to 0, P will vary from to T. Thus if T represents the 
greateat tension which the metal is capable of sustaining without im- 
pairing its elastic qualities, the pressure P of the gas in the interior of 
the bore should never ^ceed T. This principle was established by 
Lame, and afterwards followed by all other authors who have consid- 
ered the resistance of hollow cylinders. 

4. — Application to ordnance. 

The thickness of old cannon at the breech and re-inforce, that is, at 
the part exposed to the greatest tension of the powder gas, is usually 
about one caliber or twice the radius of the bore. If we introduce this 
quantity in formula (1), R^^ 3 R and P = 0.8 T. Thus in increasing the 
thickness of the metal from one caliber to infinity, the resistance of the 
gun is only increased from 0.8 T to T, that is to say, one-quarter of its 
original value. 

If a great weight of metal is to be disposed of, it is better to lengthen 
the piece, or better still to increase the caliber. In either case we at- 
tain better results from the powder and may hope to develop sufficient 
energy in the projectile to accomplish our purposes without unnecessarily 
straining the gun. 

Let us take, for example, an old smooth-bore gun of 30, of which the 
caliber is 164°^™.7, and the thickness at the re-inforce about 1.17 of the 
caliber, or 2.34 of the radius of the bore; it will require to be made of 
excellent metal to sustain a prolonged fire witli a projectile of 15 kilo- 



12 

* 
grams, and a charge of i)owder of 5 kilograms, equal to one-tliird the 
weight of the projectile. By reaming out the bore of this gun 8"^°^.2, 
that is to say -^ of the radius of the bore, the gun is converted into a 
piece of 40. If by the same process 21™™ of metal be removed, we shall 
have a piece of 60. The pressures which these three guns will sustain 
will be, respectively, 

0.82 T, 0.79 T, and 0.73 T. 

The resistance of the gun of 60 will be but ^ less than that of the 
smaller piece of 30. 

Let it be supposed that the charge remains fixed at 5 kilograms of 
I)Owder for the three guns; the guns of 40 and 60 will fire, respectively, 
spherical projectiles of 20 and 30 kilograms with normal charges of J 
and ^; the gun of 40 will present a much greater security than that of 
30, and all danger of explosion will have disappeared in the gun of 60.* 

THE RODMAN PROCESS. 

Let a gun having a thickness at the re-inforce of one caliber be taken 
for discussion. If, by any means whatever, each part of the metal in 
the whole thickness of the gun can be made to do an equal share of 

rk, the pressure that may be developed in the bore will be represented 
the formula 

r 

from which we deduce by making R = 3r, P=:2T; that is to say, that 
the resistance will be double that of a cylinder of unlimited thickness 
and cast in the ordinary manner. 

Major Eodman has realized this principle, or at least partially so, by 
a peculiar process of fabrication which has been applied to guns of large 
caliber. 

These guns are cast on a hollow core, and the mold is placed in a fur- 
nace. After the casting, a current of water circulates in the core while 
the flask is heated on the exterior. The result of this arrangement is, 
that the central parts of the casting are first cooled and solidified. After 
the whole mass becomes solid, the exterior layers are more expanded 
than those near the bore, and after the cooling they remain distended, 
and exert a compression on the inner layers, which aid the latter in re- 
sisting the pressure of the powder-gas. 

Experiment shows that the Eodman process effectively increases the 
resistance, but it is evident that its efficiency can only be partial. In 
order that the result should be complete, it would be necessary that the 

*Tlie autlior in a note fnrtlier states that it is not his intention to especially rec- 
ommend the construction of guns of large caliber. Other considerations enter into 
the construction of modern artillery. The article, however, is none the less impor- 
tant as it illustrates a useful principle. — H. B. R. 



13 

gun should cool aud solidify regularly from the interior to the exterior, 
and this cannot take place. The metal which forms the flask of the 
mold, although very much heated, must remain at a temperature lower 
than that of the casting 5 otherwise it would melt, or, if it were refrac- 
tory, dilate and detach itself from the sand, causing the mold to break 
up under the pressure of the liquid metal. It follows, therefore, that 
the layers of metal in contact with the mold will be solidified, while 
those nearer the bore will still remain in a liquid state. 

Q.— Effect of vibrations. 

If the pressure of the powder-gas is slowly developed, so that at each 
ifistant the equilibrium between this pressure and the elastic resistance 
of the different layers of metal is maintained, the work of the gas at 
each instant will be destroyed by the work of resistance due to elastic- 
ity, and the two forces will be in equilibrium. If, on the contrary, the 
pressure of the gas is suddenly developed at its maximum, and is main- 
tained for an appreciable time, the work of the gas will be in the first 
instants greater than the resistance of elasticity, and vibration will be 
produced. 

If we let i be the extension or expansion of the surface of the bore 
in the case of static equilibrium, and e half the amplitude of the vibra- 
tory movements, i and e being referred to the unit of length, ^ 4- e will 
be the greatest expansion of the bore. In the case of static equilibrium 
this expansion, i-\-e^ will be determined by a tension equal to E (i -j- e) ; 
but, according to the hypothesis, the same expansion is caused by the 
tension Ei We have, then, a right to conclude that the resistance of 
a gun is diminished in the ratio of i + e to i, since it is from the expan- 
sion of the bore that the strain on the gun may be determined. 

If the transmission of force across the walls of a tube were instanta- 
neous, the resistance of elasticity at each instant would be expressed 
by the formula (1), in which P and T are variables. Calling y this resist- 
ance and X the corresponding expansion of the bore^ referred to the 
unit of length, we shall have 

The distance passed over by, this resistance in the passage of the ex- 
pansion X to x-^-dx being EcZa?, the work developed in this small move- 
ment will be, for each unit of surface of the bore, 

and the total work for the expansion will be 



I I ydx= 



R I ydx= ^n,^2 I^E Y^ 



14 

During the same displacement of the surface of the bore, the force 
P produces a motive force equal to PE x; as this force is expressed 

we shall have 

PE X= T>/2 I T?2 ^^ ^^» 

The living force developed at the outset of the movement will be en- 
tirely extinguished when the motive force equals the total resisting 
force just found j whence it is easily seen that 

x=2i. ' 

In reality, the two hypotheses just given j the instant transmission 
of force, and the complete destruction of the living force, are not true, 
and the tubular envelope, strictly speaking, is never in equilibrium. 
The exterior surface is always a little behind the interior surface. It 
is consequently less distended than we have supposed in the formula 
(1), and the evaluation of the resisting force is a little too great. On 
the other hand, the same resisting force is a little less than the motive 
force, for when the movement of expansion is finished on the interior 
or surface of the bore it still continues on the exterior as a small portion 
of the living force. Thus in supposing equality between the two ex- 
pressions 

Wm:e^^^^^^^e-+e^^2 

two errors are committed in an inverse sense. But these errors are 
very small, and taking into consideration the rapidity with which 
motion is transmitted from the interior to the exterior of the tube, we 
may write ^ = 2i as sufficiently exact. This equation signifies that : 

The expansion of the tore of a gun produced hy a pressure suddenly de- 
veloped is double that produced by a progressive increase of the same pressure. 

In practice the pressure of the powder- gas is not developed instanta- 
neously, though rapidly, so that the value of i + e will be found be- 
tween i and 2i. 

7. — Influence of grooves. 

In a rifled gun the pressure of the gas is not upon the surface of the 
bore, but upon a cylinder tangent to the bottom of the grooves. It has 
been shown how little the resistance is diminished by increasing the 
bore by a small quantity j now,'^in a gun of a caliber of 16 centimes the 
depth of the grooves does not exceed the 0.073 of the radius of the bore, 
and for other guns it is still less. In such a case but a small error will 
be committed in neglecting the depth of the grooves. But there exists 
in guns deeply grooved another cause of destruction. This is the effect 



15 

of the powder gas wMch enlarges the grooves by compressing the 
lands which separate them, causing the sides of the groove to separate 
from the bottom, and giving rise to cracks and fissures in the re-enter- 
ing angle formed by the junction of the surfaces. This cause of de- 
terioration is rapid and has been very marked in many systems of rifled 
ordnance. Hence the necessity of diminishing as much as possible the 
depth of the grooves and joining the principal lines of their profile by 

means of curves. 

8. — Effect of guns becoming heated. 

Many persons regard the elevation of temperature resulting from a 
rapid and prolonged fire as one of the causes of the bursting of guns. 
^N^othing authorizes this presumption j on the contrary, it may be said 
that the layers of metal which are near the bore become exijanded by 
the heat more than those that are situated near the exterior, distending 
the latter, which compress the inner layers and render the danger of 
rupture less imminent. 

The lowering of the modulus of elasticity also acts in the same way. 

9. — Observations relative to permanent strains. 

When the interior layers of a metallic tube are submitted to a tension 
greater than their limit of elasticity, the strained parts which remain 
elastic are alone proportional to this tension, the strains are permanent 
and increase, consequently the exterior layers are more distended than 
they would have been without the intervention of the permanent strains, 
and thus support the interior layers. It results from this, that formula 
(1) gives in this case, too small a value to P, and that the difference be- 
tween these values and the actual resistance of the tube increases as the 
permanent strains are greater. 

Thus two metals may equally resist a given tension ; one remains 
perfectly elastic, the other undergoes a permanent strain j and if they 
are used in the fabrication of two equal tubes, the tube made of the 
first metal will have a resistance inferior to that of the other, and the 
superiority of the former will be more marked as the permanent strain 
is the greater. These considerations explain the great resistance of old 
bronze cannon as compared with those made of cast iron. 

Resistance of old smooth-hore guns. 

It was found by comparing the initial velocities of projectiles and the 
the velocities of recoil furnished by guns of different lengths of bore, 
that an empirical formula might be established, by the aid of which 
the pressure of the gas in the bore might be computed within a certain 
degree of approximation. This formula applied to a gun of 30, throw- 
ing a projectile of 15 kilograms, with a charge of J, gives 830 kilograms 
for each square centimeter as the pressure of the gas. 

If we introduce this quantity in formula (2) that we shall find further 
on, and make W= 3.17 E, we find T = 10 kilograms per square milli- 
meter. ' 



A 



CHAPTEE II.— BUILT-FP TUBES AND GUNS. 

1. Co7iditions of resistance of an elementary tube. 

A built-up tube is composed of several simple tubes placed concen- 
trically one over the other. These tubes are called the elementary tubes. 

A built-up gun may be regarded a built-up tube. In order to make 
a built-up tube, the elementary tubes are put together successively from 
the center to the circumference; each one of them is reamed out to a little 
smaller diameter than is necessary for its adjustment. The tube is 
placed after its expansion by heat so that each one in cooling exercises 
a strong compression on the inner one next it. It results from this ar- 
rangement that the elementary tubes are tangentially compressed next 
the bore of the gun, while those on the exterior are distended. 

When the entire system is submitted to an interior pressure the 
compression of the elementary tubes diminishes, disappears even, and 
sometimes changes to expansion if the interior pressure is sufficiently 
raised, while the exterior tubes, already distended, are still more strained. 

The exterior tube has one of its two cylindrical surfaces free, and 
follows the law of homogeneous tubes subjected only to an interior 
pressure, while all the other tubes are subjected to two pressures, the 
one interior and the other exterior. The formulas which relate to the 
latter are deduced from the two equations (b) of Chapter I, Part II, as 
follows : 

(T+P)R2=(T'4.p/)R/2, T-P=T'-P'- 

eliminating T', we have 

m T-p^Ljt^'-p/^^E^ 

^^ W^-n^ B'^-W 

Let one of the elementary tubes submitted to two pressures be con- 
sidered. 

If t be called the tension of the metal in the direction of the circum- 
ference of a point situated at a distance p from the axis, and p the press- 
ure acting on the same point in the direction of the radius, we shall 
have Chapter I, Part II, equations {a) and (c) 

(T+P)R2=(^-fj)>2 T-P=*-i> 

{t+p)p^={T'+-p')W' t-'p=T-V\ 

16 



17 

Eliminatiiig T, T', iiiid j; from these four equations, we have 

R^ n-+p^ R- E^+/ ^- " f; -^ '^ 



2+1 ",+1 

'R^ R^2 



IP 



Differentiating, we have 

If P is smaller than P', the formula (2) shows that T is negative, 
that is to say, that the interior snrface is compressed; ^^ is then positive; 

consequently the algebraic value of t increases with ,o, which sionities 
that the compression goes on diminishing from the interior to the 
exterior of the tube. 

If we make />=R', then t has a negative value, which signifies that 
the whole thickness of the tube is in a state of compression. 

If P is equal to P^, T is still negative, but ^ reduces to zero, what- 
ever value may be given to ^ -, in other words, all the circular fibers are 
equally compressed. 

2 P' R/2 

If P is greater than P', but smaller than p/2T:p2? T is still negative, 

as is T ; there is still compression of the interior surfa<^e; but this 

compression increases from the interior to the exterior of the tube, 
where it attains its maximum. 

2 R'^ 

Finally, if P is greater than P^ ^ 09, T is positive, the interior 

K^ -{- R~ 

surface is expanded; but- is negative, and it follows that the ten- 

dp 

sion of the fibers diminishes as the distance from the interior surface 
increases. 

It results, therefore, from this discussion, that the maximum tension 
of the fibers (if there is any tension) is always on the interior surface; 
consequently it is necessary, in order that the elementary tube shall 
withstand the interior pressure, that T shall never exceed the tensile 
strength of the metal of which the tube is formed. For the sake of 
brevity, in speaking of the tension and limit of elasticity of a tube, the 
tension and limit of elasticity of its interior surface will be undeistood. 
5318 2 



\ ^ 



18 

Calculation of the resistance of a hiiilt-iq) tube. 

Let Eo be the radius of the bore of a built-up tube, Ej, E,, E3 . . . E,„_i 
the radii of the surtaces of the contact of the several elementary tubes 
and E^ the exterior radius ; Po, Pi, P2 . . . P^, the pressures which act 
normally to the cylindrical surfaces whose radii are Eo, E^, . . . E„ ; 
Po is the pressure which the tiibe resists, and is the quantity to be de- 
termined. In the general case P„, is the atmospheric pressure and will 
be neglected. 

Let us also call To, Ti, T2 . . . T^_i, the tensions of the circular fibers 
composing the interior surfaces ofthe elementary tubes, and To', T/jTz' . . . 
T'^_i the tensions of the exterior surfaces of the same tubes. A built- 
up tube will support the greatest i)ressure when all the elementary 
tubes are strained at the same time to tlieir limits of elasticity respect- 
ively. Suppose this condition to be fulfilled, Tq, Ti, T2 . . . T^_i will 
represent this limit for the tubes respectively corresponding to these 
tensions. This being granted, and considering the two general equa- 
tions 

(T-fP)E2=(T'+p/)E'2, T-P=T'-P'5 

we deduce, eliminating T' and solving with reference to P, 






This formula applied to a tube occuj)ying a place in order of 11 + 1, 
becomes 



By making n=m and P^=0, we obtain P^_i. ]S"ow, making n=r/i— 1, 
we shall have P;«_2. Knowing P^_2, T^_3, E^_3 and E«_2, P^_3 can be de- 
termined in the same manner. In continuing thus, we shall finally ar- 
rive at Po, which will be the resistance of the whole system, represented 
by the greatest pressure that it will bear with safety. 

It may be remarked that: 

rt •p2 T>2 "f>2 

In making this substitution in formula (3), it becomes after reduction : 

This formula is more convenient for practical work. 

Eeferring to formula (3), if ve make 7i=0 and Pi=0, it becomes: 



E^o 
E^ + E^o 



Po— To-p2 , T>2 } 



19 

P<5 represents the resistance of the central tnbe alone and deprived of 
the supiiort of the exterior tnbes. 

If we make )i = and not Pi, we have: 



biit we have, niakino- n 



3 _T ^" 

- J-O p2 




) 




=1, 








P.=T,g 




I 


2E22 



Substituting this valne in the expression for Po, we have 

Jr-O-lo JJ2^_^J^2^+ J-l JJ2^_^iJ2^ R2^_^1^2^ •+--^2^^4 E^, E^^ + EV 

an expression in which the first term represents the resistance of the 
central tnbe, the second the fraction of the total resistance furnifched by 
the second tnbe, and the thii^d the fraction of this resistanc^^ to all 
the tubes exterior to the second. II 

Eeplacing P2 by its valne as a function of T2, of P3, of E2 and of E3, 
and continuing in the same way, we shall have a development of Pq in 
which each term will represent the part of the resistance furnished by 
one of the tubes of a built-up gun: 

. p _^ E-\-E\, E^^-E^ _^E^_ 

^'^^ • . • • ^0--L0j^2^_^J^2_+J-l_^2^_^Ji2^ E\ + E'o 

^rr. Fv^3-E\ 2E^ 2E^2 

+ ^^E^34-E-2 E-;+K% E22+E^i 

^^ E ^4-E^. 2R^ 2E^2 2E^3 

-^-^^E^4+E^'3 E^o+E-1 E^2+E^ E23+E\ 



o. E2,.^-E2„ 2E^i m\_ ^E^+i^ 

"E^.+i+E^ E^o+E^ E^+E^s • • • E^.+i+E2"n 

^^ 2E^ 2W, 2E^+i 



E^o+E^ E^-hE^'2 E^+E^„+l 

§ S.—EXA^riPLES OF APPtlCATIOW v, 

1. — Cast-iron guns Iwojyed icitJt steel. ( 

Consideration of a cast-iron gun having the reinforce strengthened 
•with hoops of puddled steel. From experiments made at the foundries 
of ]S"ever.s and the forges at Guerigny, it was found that bar puddled steel 
of the sa!ne quality as used for guns had a tensile strength of about 
forty kilo-rams per square millimeter without contracting a permanent 
elongation, and broke with a load of about 55 kilograms 5 but made in 



20 

the form of a tube for a gun, the same metal had only an elastic resist- 
ance of from 2;") to 30 kilograms. The lower figures will be adopted^ 
say 25 kilograms. 

As for the cast iron we will suppose its resistance reduced to 8 kilo- 
grams per square millimeter, in this particular case, in order to take 
into consideration the influence of the grooves and other causes of 
deterioration, which may result from its immediate contact with the 
powder-gas and with the projectile. This being understood, let it be 
granted that the total thickness of the gun is one and a half calibers, a 
caliber of cast iron and half a caliber of steel. 

We have from these hypotheses : 

Ej = 3Ko; andE2 = 4Eo. 

The pressure, l*i, exerted b^^ the hoops on the cast iron, at the moment 
when the powder-gas develops the greatest pressure, is giA^en in for- 
mula (1), which in the case before us becomes: 

_^z 32 

Pi=Y2^-r--o2x25=7 kilograms per square millimeter, 

and the interior pressure by formula (4) becomes 

32 I 

P(i=o2_r'] X (^^+^) + ^=19 kilograms per square millimeter, 

or, after the mode of expression in general use, 1900 kilograms per 
square centimeter. This represents the resistance of the gun. 

A gun made entirely of cast iron, cast in a single piece, would have 
a resistance equal to: 

42— 1 

vg-j-yX 8=705 kilograms per square centimeter. 

The resistance of a cast-steel gun, allowing twenty kilograms per milli- 
meter for the strength of this last metal, will be 

42— 1 

n_r^X 20=1760 kilograms. 

Sixteen centimeter guns of cast iron reinforced loith steel lioops. 

The guns of 16 centimeters are made from a single piece of cast iron, 
forming the body of the gun, and a row of hoops of puddled steel ex- 
tending to the trunnions. The tests to which the hoops were subjected, 
before being received, insured the malleability of the metal but not its 
elasticity. Thus the guns of this model which have been last fabricated 
have less durability than their predecessors. 

As a secondary cause of the inferiority of this model of gun, the 
depth of the grooves must be considered ; this is somewhat lessened by 
the form of a flat arch being given to the profile of the grooves so that 
there are no re-entering angles. 



21 

The transverse diuieiisions of the gnu are as follows : exterior di- 
ameter of the hoops 584 millimeters, diameter of the reinforce under 
the hoops 472 millimeters, caliber 164.7 millimeters. 

If we use these data iif the first example already given, and if taking 
into consideration the want of elasticity of the hoops, and calling the 
tensile strength 20 kilograms per square millimeter, the cast iron hav- 
ing a tensile strength of 8 kilograms, we find 1360 kilograms per cent- 
imeter for the resistance of the gun. These guns were constructed to 
fire a shell of 31 kilograms, witli an ordinary charge of cannon powder 
-3.5 kilograms, with a wad between the charge and the projectile. They 
stand this fire, jbut in increasing the charge to 7.5 kilograms, with a pro- 
jectile of 45 kilograms, the guns have always burst after a few rounds. 

2. — Cast-iron gun^ lined icith steel tubes — Falliser system. 

Let a'cast-iron gun pros ided with an interior steel tube be now con- 
sidered. 

As in this case the cast iron is not directly submitted to the same 
causes of deterioration as exist in the preceding case, a tensile strength, 
of 10 kilograms may be assumed. For inverse reasons the tensile strength 
of the steel will be lowered. 

Cast steel, which alone is employed for the manufacture of the central 
tube, is often full of bubbles or air spaces. Hammering may cause these 
flaws to disappear, but only in closing up the sides, so that the resist- 
ance of the metal is not increased. It is possible by certain processes 
of fabrication, such as casting the metal under a pressure, or using a 
lieavy sinking head, to almost entirely avoid these causes of weakness ; 
but it is necessary to avoid erring on the other side in attributing to 
<;ast-steel tubes a less resistance than to hoops of puddled steel, which 
are made from masses of metal relatively weaker. 

For these several reasons, we will fix on 20 kilograms per square 
millimeter as the tension to which cast-steel tubes should be submitted. 
This strength of 20 kilograms corresponds, besides to the limit of 
elasticity of low steel. 

AVe will suppose, as in the preceding example, that the gun is a 
^caliber and a half thick, and one caliber is of cast iron, and half a caliber 
of steel 

From the data given, Ei will be equal to 2Ro. and E^ to 4Eo; we shall 
c-onsequently have 

Pi=4^^i-2Xl0=6kil. 
and 

Po=^^(20+ 6) + 6=21^6 
per square millimeter, or 2160 kilograms per square centimeter. If we 



22 

had fixed, as in the preceding example, 8 kilograms for the tensile? 
strength of the cast iroD, we should have found: 

Pi=4\3 and Po=18K90. 

This value of Pq is about equal to that found in the case of the hooped 
gun. 

Thus regarding only the elastic tension, we recognize that the tubed 
gun (Palliser system), for the same thickness of steel, presents about 
the same resistance as the hooped gun. But the elastic tension is not 
the only one to consider. The bore of the gun is liable to the accidents 
of discharge, which may determine exaggerated pressures. In these cir- 
cumstances cast iron, when forming the bore, soon reaches the degree of 
tension which determines rupture, and cracks are formed which increase 
the surface of gas pressure, and as the gun is again discliarged, sooner 
or later the piece is destroyed. 

Steel, on the contrary, employed under the same conditions, ^¥ill break 
with difficulty under a pressure of 55 kilograms per square millimeter,, 
and the limit of elasticity m^y be greatly exceeded without breaking j 
thus the chances of accident will be much diminished. From these con- 
siderations the tubed guns (Palliser) are to be preferred rather than 
those of the hooped system. 

3. — Guns hooped and tiibed. 

Let us again suppose a cast-iron gun to have walls of a caliber and a- 
half in thickness, of which half a caliber is cast iron, and in which this^ 
metal is placed between the exterior and interior surfaces. 

The hoops are to be two-thirds of the radius of the bore in thickness^ 
and the central tube one-third of this same radius. 

We shall then have : 

Kow supposing that the tension of the cast iron is limited to 8 kilo- 
grams per square millimeter, that of the hoops to 25 kilograms and 
that of the tube to 20 kilograms, we shall have 

122—102 

2 — 122 -I- 102^" "^^^ 



from which 



102—42 

i*i=ro2i:4-2(S+^-5i)+4-5i=i^-^'5 

Po=42~32(20-fl3.57) + 13.57=22.97. 



Thus the gun will resist a pressure of 22'^97 per square millimeter or 
of 2297 kilograms per square centimeter. It will consequently have 
more resistance than the gun in the preceding example. 

If we give a tensile strength of 10 kilograms per square millimeter to 
the cast iron, we shall find for Po 2483 kilograms per square centimeter,, 
and the advantage of the mode of construction in whicli tubing is com- 
bined with hoopifig becomes more evident. 

>4 



•23 



§ 4.— Conditions of the iiia\iiuiini ol* resi^iitance. 

Ill the preceding examples we have arbitrarily given the thickness 
of the two or three concentric tubes, which constitute the reinforce of a 
gun, taking as a guide the practical considerations drawn from the na- 
ture of the metals employed. Xotably. in relation to cast iron, the fol- 
lowing conclusions have been reached : 

1. That in a gun of this metal, simply strengthened by the addition 
of hoops or of a steel tube, the mass of steel should be but a small i>art 
of the whole mass of the piece, or the economy which is to be kept in 
view by the combined employment of the two metals will be lost. 

2. That cast iron, on account of its brittleness. should only be used in 
large masses of considerable thickness. 

3. That this thickness is necessary in order that the gun may resist 
transverse strains proceeding from the exterior or resulting from the 
discharge. 

4. Finally, it is necessary to increase the resistance which the circu- 
lar part of the body of the gun opposes to the pressure of the gas at the 
bottom of the bore, since we are ignorant how mucli the compressive 
energy of the metal in the direction of the radius will act upon the re- 
sistance. It would not be without interest to know, theoretically, the 
division of the reinforce which will give the greatest strength to the. 
gnn. 

And in general to know what relations of this kind it is necessary r* • 
establish between the different elementary tubes in order that the giin 
may attain its maximum resistance, having given the number, the 
nature of the elementary tubes, their order of superposition, as well as 
the exterior and interior radiiof the built-up tube itselt^ let us imagine 
then a tube composed of any number of simple concentric tubes, and 
consider two of the latter occupying any position whatever, but follow- 
ing one the other immediately in the order of superposition. Let us 
call: 

i\ the interior radius uf the smaller of the two elementury or 
simple tubes, 

R. the exterior radius of the greater, 

p. the radius of the surface of junction of the two tubes. 

P. the normal pressure on the surface whose radius is r. 

P'. the pressure exerted on the surface whose radius is K. 

T, the limit of elasticity of the interior tube, 

T', that of the exterior tube. 
Let it be admitted that all the radii remain constctnt. witli the exce^)- 
tion of ,o, which we will consider as variable. The pressure P is repre- 
sented by the formula: 



(See formula (5).) 



24 

V is coustant because it depends solely on the limit of elasticity and 
the radii of the elementaiy tubes, which are exterior to the two under 
consideration, and by the hyi^othesis these quantities do not vary. R 
and r are given, and P is the function of the only variable, p. 

The pressure Po, which measures the resistance of the gun, augments 
and diminishes with P, and only with P, while the limits of elasticity 
and the radii of the tubes which are interior to the radius r remain con- 
stant; Po will only be a maximum when P is at its maximum. We will 
now seek the value of />, for which P is a maximum. 

Reducing the terms of P to the same denominator and adding them, 
we have 

' p-/ ( T-2T0 + />^[(B^-r'^) T + 2B^T^ + 4R^ P^] -W r' T 

The differential of P after reduction gives the following equation : 

P^ [2W P' + (2R2 + r2) T' - r2 T] - 2/>2 R2 r^ (T - T') 

- R^ r2 (2P' -f T + T') = 0. 



Putting 



we find 



2R2 P ^+ (2R2 + r^) T^ — r2 T = A, 
K2r2(T — TO = B, 

R4f2(2p/4-T + T')=C, 



^ A 



C is always positive, since P' is a quantity essentially positive and 
T and T' represent positive tensions. If A is equally positive, the prod- 
uct AC will be positive, and the radical will give the sign to the value 
of ^'^5 //- cannot then be negative, and the radical is to be taken with 
the sign +. 

If A is negative, B will give its sign to the numerator of p'^] but B 
will in this case necessarily be positive, and A cannot be negative un- 
less T is greater than T'; p^ will then be negative and ^o imaginary; 
then, if there is a maximum, A is positive, and from what has preceded 
the radical should receive the sign +. 

Thus in all the cases in which there exists a value of p which renders 
P a maximum the square of /> is given by the formula: 



(7) P'= 



B+VB'+AO 



If the two tubes under consideration have the same limit of elasticity, 
whivih occurs in treating of two rows of hoops or two superposed steel 



25 

Tubes, formula (7) becomes more simple since T = T^ which introduced 
in the values of A. B and C, gives 

s o- = llr. 



If there are more than two consecutive elementary tubes, having the 
same limit of elasticity, any two of them whatever, if next to each 
other, should satisfy formula (8), from whence the following practical 
rule : 

In a tube built uj) comecutively with elementary tubes of the same metal^ 
or icith tubes of different metals but having the same elastic resistance, the 
raflii of the surfaces of contact of the elementary tubes should increase in 
{fco metrical jjrot/ress ion from the interior of the smallest to the e.rterior of 
the largest tube. 

If a built-up tube is formed from a series of simple concentric tubes 
oi the same metal, of which the radii have been determined by this 
rule, the calculation of the resistance becomes easy and reduces to the 
summation of a geometrical progression. 'Formula (5).l 

If we wish to find the values of p. which will make a maximum, Urst 
of the sum of tlje first two terms of P, second the third term, formula 
'6 1, it will suffice to introduce successively in A. B and C the following 
hypothesis: first. P' = 0: second. T=0. and T'=0. 

It will be found that the values of o^, which correspond to the two 
maxima, are. for the sum of the two first terms of P. 



(9) . . . ,- = "' 
-and for the third term. 



T_T/^+E-/V^T'[E-(T-hT^)^+r^(T -T : 



Frequently, with a sufficiently close approximation, we may neglect 
,-^(T'— T) by the side of Pv-(T' + Ti and 2R-T\ and replace T + T^ by 2T': 
we shall then have for the two first terms as for the third. 

y- = ~Rr. 

Thus the sum of the two first terms of P formnla (G) often becomes a 
maximum for a value of .o~, which difiers little from the product Pr, 
while the maximum of the third term always exactly corresponds to 
o-=Jir. This circumstance gives, then, a greater practical utility to 
the rule already given, and often permir> irs extension even to element- 
ary tubes of ditferent elasticities. 

Eeturning to the case when it is wished to determine exactly the re- 
lations of thickness giving the maximum resistance. 

Formula (0) applies to the case of a tube composed only of two ele- 
mentary tubes of different metals. If there are more than two element- 
ary tubes, or if of difierent metals, or if they have not the same limit 
of elasticity, it will be necessary to resort to formula (7), and to give to 



26 

the elementaiy tubes arbitrary thicknesses, which can afterwards be 
corrected by successive substitutions. (See third example, further on.) 
A division of the tube, established from the rule already given, may 
be taken for the calculation, this rule giving, as has just been exi)lained^ 
a first approximation, even when the limits of elasticity of the metals 
employed differ notably among themselves. 



t 



§ 5. Practical examples. 
1. — Hooped cast-iron guns. 



Let us seek the interior radius of the hoops corresponding to the aiax- 
imum of resistance of cast-iron guns hooped with steel, and suppose that 
the total thickness of the reinforce is a caliber and a half. We will fix 
the tensile strength of the cast iron at 8 kilograms and that of the hoops 
at 25 kilograms per square millimeter. 

Making, in formula (9), 

T=8, T'=25, R=4r, 

we shall find 



hence 



,o^=2.9r%- 
p = 1.7r. 
From this we deduce the thickness of the cast iron 

1.7/-_r= 0.7r 

and that of the steel 

4r — 1.7r=2.3r. 

The value of P, or the interior pressure measuring the resistance of the 
gun will be raised to 3270 kilograms per square centimeter. 

But the thin cast-iron tube will not practically fulfill these conditions. 

It is not possible, then, in this particular case to admit the relation of 
thickness between the cast iron and the steel, which would oppose ttie 
maximum of resistance to longitudinal fracture; but we arrive at the 
same time at this curious result, that the resistance of a cannon would 
be augmented in giving it an interior liniug of about 0.7 the radius of 
the bore. 

2. — Cast-iron guns icUh steel tubes. 

Let us pass to the opposite system of construction, that of a cast-ii'on 
gun lined with a steel tube, and having, as in the preceding case, a 
caliber and a half of thickness. Bearing in mind what has already been 
said on the nature of the two metals, we will take as the limit of tensile 



27 

strength 20 kilograms per square millimeter for tlie steel tube and lCt> 
kilograms for the bodj^ of the cast-iron gun. 
In formula (9) making 



T=20, T' = 10, E=4r, 

^•^=5.52 r2; 



we find 

whence 

P=2.35r. 

The thickness of the steel will be 

p — r=1.35r, 
that of the cast iron 

E-,o = 1.65r 

and the pressure, P or Po=2210 kilograms per square centimeter. 

Here the thickness of a cast-iron tube, joined to its large diameter^ 
will give sufticient resistance for all causes of destruction that may 
occur, and which are partly supported by the central tube. But it may- 
be remarked that the maximum pressure that we have just found, 2210 
kilograms, differs very little from that of 2160 already obtained in a pre- 
vious example, by fixing a priori the thickness of tlie steel tube at half a 
caliber ; from whence it results that t.his last thickness is to be preferred,, 
since it permits the realization of a certain economy of construction, and 
of notably increasing the envelope of cast iron. 

3. — (thhs^ both hooped and tubed. 

Let us consider a cast-iron gun strengthened by an exterior row of 
puddled-steel hoops, and with an interior tube of cast steel, and deter- 
mine the iuterior radius of the hoops and the exterior radius of the- 
central tube, so that the maximum resistance may be realized. 

We will supjjose a total thickness of one and a half calibers, the steel 
to have a tensile strength of 20 kilograms for the cast metal, and for 25 
kilograms for the hoops; the cast iron to have a tensile strength of 10 
kilograms. In this case, as there are more than two tubes, foruuila 7 
must be employed. 

Let the reinforce at first be divided as though the elementary tubes 
were of the same metal, and let the radius of the bore be taken as^ 
unity. 

From the method of division adopted as a point of departure, Ki, ex- 
terior radiu-s of the central tube, will be equal to >yi = 1.59. Starting^ 
with this number, we can determine the A^alues which approximate more 
and more closely to this radius Ej and to E2, the interior radius of tlie 
hoops. 



28 

First operation. — Calculation of Bi. 

The given values are : 

r =Ei= 1.59, whence r"^ = 2.53, 
E =E.>= 4, whence E2 = 16, 
P'=P3= 0, 
T = Ti = 10, 
T' = T, = 25. 

^Substituting in the values for A, B and C, we find 

A = 838, B = -607, = 22.680. 

Using these vahies of A, B and 0, in formula (7), we have 
^/2 — 4.55^ whence p = E2 = 2.13. 
Second operation. — Calculation of Ri, 
The given values are : 





r^ = E% = l, 




E^ = E^ =4.55, 




T =To =20, 


* 


T' = Ti = 10, 






Introducing as before in the values of A, B and C, Ave have 


whence, 


A = 207, B = 45.5, C = 1.196, 
;r.2=2.65 and /. = Ei = 1.63. 




Third operation. — Calculation ofR^- 


r2 = 


2.65, E^ = 16, T = 10, T' = 25, P^ = 0, 


whence. 


A = 839.5, B = - (jm, C = 2.373 



:and by formula (7), 

^,2 = 4.71, . ^o=:E, = 2.13. 

Falling back on the value of E2, already- found to within nearly 0.01, 
the values of Ei and of E2 will consequently be determined with a 
sufficient degree of approximation. 

Thus the resistance of a gun will be carried to its maximum when 
radius Ei = 1.63 Eo, and radius E2 = 2.13 Eq. The thickness of the three 
•elementary tubes will then be : 

the central steel tube 0.63 Eo, 
cast-iron body 0.50 Eo, 

the hoops 1.87 Eq. 



29 

The resistance of tlie gun, or Po, will be 4.")4:!) kilograms per s(iiiare 
ceiitiineter. 

A constructiou like this, however, is not admissible, since the cast- 
iron body would not be as thick as the central tube, while the hoops 
would have such dimensions as would require them to be regarded as^ 
the principal element of the gun. 

4. — Malleable tube irifhout reshtance plaeed in a resistine/ (jun. 

AVe may conceive of a homogenous gun that should receive a central 
tube oidy designed to transmit pressure. It is ])roposed to discuss the 
relation between tlie resistance of the' [)iece and the thickness of this- 
interior lining. 

The tube may possess an elastic comi)ressibility vsuflicient to support 
the compression due to the initial tension of its envelope, but without 
l)Ower to resist the least eftbrt of tension; or rather we may imagine^ 
that it possesses extensibility and malleability in an unlimited degree, 
that it follows all the movements of expansion and contraction in the 
body of the cannon, but without opposing the slightest reaction. 

In these two cases we shall have the value of //, which corresponds to 
the maximum of resistance, by making in formula (9) T=0. 

If it is granted further that R be made equal to 4r, we shall find 

^o = 1.53r. 

Consequent!}' the thickness of the tube will be 0.53r, and the value of 

Po=1.04 T'. 

Thus, by the addition of a central malleable tube transmitting the press- 
ure of gas to the resisting walls, the gun acquires a greater resistance 
than if its thickness had been unlimited, and the thickness necessary 
to give such a tube would be about one-quarter of a caliber. 

o. — Cast-iron or steel guns with bronze tubes. 

Bronze may be employed as the malleable metal in the illustration just 
given, but this metal having a resistance of its own, it will be necessary 
to take it into account. 

Let us imagine a cast-iron gun with an interior bronze tube, and see 
what the thickness of this tube must be, so that the resistance shall be 
the greatest possible : let us sui^pose, besides, 

E = 4r. 

Bronze may be considered either as elastic or malleable. In the first 
case it should not be submitted to a tension greater than 2 kilograms per 
square millimeter ; in the second case the use of old bronze ordnance 
shows that it supports heavy strains under the same dimensions, as 
well as cast iron, and that it can withstand a tension of 10 kilograms. 



30 

In the first case, considering the use of bronze within its limit of elas- 
t:icity, if we make 

T=2andT'=10, 
formula (9) gives 

/o=1.65 r. 

The thickness of the tube will then be 

O.Go. /■; 
and taking for the calculation of resistance 

Eo=r, Ei=/>=1.65r, E2=E=4r, T=2 kil., T'=10 kil. 

> 

we find 1124 kilograms for the resistance of the gun. 
If, considering the bronze as a malleable metal, we make 

T=:10 and T^=10, 
formula (S) will give 

The thickness of the tube should be taken at half a caliber. The cal- 
culation of the resistance will give 1560 kilograms per square centimeter. 

We see that these results are verj* inferior to those obtained from 
^teel hoops or tubes. 

If, instead of a cast-iron envelope, we conceive of a steel envelope, we 
^hall find 

1.36r 

for the thickness to be given to a bronze tube, and 

2535 kilograms 

for the resistance of the piece. 

Dutch system of conversion. 

The system of construction which consists in strengthening cast-iron 
guns with bronze tubes has been realized in Holland with good results. 
The old guns were reamed out and filled in with bronze castings,wliicli 
were bored out to the proper caliber. 

6. — Bronze guns ivith steel tubes. 

It must be supposed in this case that the steel tube possesses a cer- 
tain elasticity. It is necessary that the bronze should not be strained 
above its limit of elasticity, because if it were it would separate from 
the interior tube when the latter contracted after the explosion. Con- 
sequently, in formula (9) we will put 

T=20 and T'=2. 
If we again suppose 

E=4r, 
we shall find 

P=4rMr', 



31 

that is to say, a value of p greater tbau R ; which signifies that there is 
no value of p which gives a maximum, and that the gun should be con- 
structed entirely of steel. 

The combination which we have just examined is not, then, a suitable 
one as far as the resistance is concerned. It was nevertheless attempted 
in the construction of a gun at Gavre; the promjit destruction, how- 
evar, of the piece, of which the tube was of good Krupp steel, confirmed 
the indications of the theory. 

Exx)eriments have lately been renewed in regard to tubing bronze 
guns. The end to be attained was to give greater hardness to the sur- 
face of the bore, and to jDrolong by this means the life of the piece. 
The difference between these new efforts and former attempts consists* 
III the substitution of steel for iron. It is important that the steel tube 
sliould be malleable and preserve permanently the extension which it 
acquires under the pressure of the powder-gas, otherwise it will sepa- 
rate from the bronze. 



C(MA. 



CJLrx/^n^ 



iLc Ou/muj^ ^' ^ 



1 



?• 



.'H.I 



C<iC - 







Longitudinal and Hoop Tension in a Thick Hollow Cylinder. 
By Lieut. C. A. Stone, U. S. N. 



Consider a thick hollow cylinder, closed at both ends and pressed from 
within by a pressure P, and from without by a pressure P' . , The notation 
being the same as that given on page 6 of Virgile, we have, for a longitudi- 
nal section through the axis: 

tdp = PR — P' R\ (1) 

R 
for a cross-section, we have 

•^' ^ PR'—P'R'^ ,^. 

spdp = r —^ (2) 

R ^ 

where s is the longitudinal tension. 

Let t =f' [p): then from (1), 

t=:f'{p)=-p-p^^, m 

dp 
Let s = — ; then from (2) , 



/; 

. we 

/; 



From (3) and (4), 
t 

and 2s — t=:—p. (6) 

dt) 
First: let f = s; then from (5) -^ = ; .^ — p = m, m being a cc 

From (3) and (4). we have also, ^ =sz=??i. This satisfies (1) and (2). 

For the second sol 

(3) and (4) , we have 



^ '- 2- dp' ^'^ 

dt) 
First: let f = s; then from (5) -^ = 0\ r. —p=^m. m being a constant-, 

s = m. This satisfies ( 

For the second solution, let f = — s in (5), then t =z -7-" and. from 

4 dp 



3p dp dp 4 dp 



(7) 



"^ 2* dp' • • p ~ 3' p ^ 

from which we see by integration that p is of the form ~^. 

Differentiating, we have, 

dp ^ _a 

'^~~ 3-~|' 

whence t = -L.^^^^. (8) 

4 dp 3^^ 

We have also P = — -- and P' = -^ri • (9 ) 

Rj -K 3 

Substituting these values in (1), we have 
R' 



%=PR-P'R' = arj: L_-|, 



(10) 



RP' 

which is thus satisfied. 
Combining the partial equations, we have, 



t = m + -^, (11) 

op's 



a 



(12) 



p — —m^ J-; (13) 

pz 

.\ St--p = 4:m, whence 3 T— P=i^T' — P' (14) 

Act, 4 4 

and ^+i>=:~^, whence (r 4- -P)i^' = (^' +P')i^'^ (15) 

Sp-i 

P P' 

Eliminating T' = T — -f- -^, we have 

o o 

S E^ {T -^ P) =^B'^ [Q T -- P + 4: P') 

••• T=-^ 4 4^ • 16) 

From (12) nnd (13), we have 

s-|-p = -^, whence {P+S) E^ z=z {P' -\- S' ) R'^ (17) 

and 3 s H-_p — 2m, whence 3 Sf -j- i? = 3 S'' + P', (18) 

4 4 / P 2 P' \ 

whence P"^ (p_^ g) = P'^ V'^' H" ^ "^ "T^j ' 

^.^^^P(3Pf-P-t)-2P-pl 

3(P'|-Pf) • ^ 

Where S, and 5^' are values of s for p=:R and p-=z R\ 
From (17) and (18), we have 

^l(s+p)=:pt(.g + P) 
and 3s + P = BSf-hP 

.♦. p3 (3^_2.s-}-P)=P^^P-j-Sr) 

^^ S (3 p^ — P^) -h P (p*— P^) = 2 s p^ 

— ^ ^^ ^ ~ -^!}jf.:P_(/^^— -^^) 

Substituting the value of aS' from (19), and reducing, we have 

P E^ (3 p^ — R' h — P' R'~hsp'^—E^) 

ds __ 4PJ R'Up—P') 
whence -j ^^ ~ —- 

^'^ 9p3 (P'3_^/f) 

and R' ^ R f ^ ^® ~^' therefore the longitudinal tension is greatest on the 

outside. 

Let p = R' in (20) 

2PP3— P'(3P'3~P3 ) 



S": 



(21) 



3(P'-^— P«) 
From (14) and (15), we have 

3^— p=:3r— P 

and p^ {t-\-p)=zR^{T-j-P) 

4 4 4 4 

r (3 o"^4-p3) — p (p'^ — pn 

whence t= ^' ' ^ ^ . (22) 

4p3 

Substituting the value of Tfrom (16), we have, after reduction, 



P R^ (3 p^ + R'h - P^ ^^ (3 P^+R"^ ) .._. 

3p3 (^'3 _^3) 



Differentiating 






^^ /^ 3~ is negative; therefore the greatest hoop-tension is on the in- 



If P > P' 
and R > 

side, where p= P; and its value T is given by equation (16). 
To determine m: from (14) 

substituting the value of Tirom (16), and reducing 

PR^ — P' R'^ 

m = 1 ^— • (24) 

R'^—R^ 
To determine a: from (11), after substitution of particular values 

a = dR^{T—m) 

R^ R'^ [P—P') 
whence a=z ^ ^ , (25) 

R'^ —R^ 

Example : 

1 r ,7. 19 

Let P = 1 j I 45 ^ 

P' = 8 . Qvi/i Tvro ViC^^cx ! 5^'= — P 



1 and we have \^ — 77 
1 13 

l^ = -45^ 



and P' =: 

»S being a longitudinal compression, since it is negative. If the longitudinal 
tension were uniform over the area of the cross -section, its value would be 

P 

63- 

From equation (20). we see that, when P^ = and Sp^ =:P^%orp = .489P^ 
about. 5 = 0; which shows the position of what may be called the neutral 
surface, within whic^there is a longitudinal compression, and without which 

a longitudinal tension. In (19), when 3 P^ < P^^^ g jg negative, showing a 
longitudinal compression. Making P^ = P in (20), we find the longitudinal 
tension s is uniform and = — P, as might be expected. 
From (20), making s = 0, we find, for the position of the neutral surface, 

p^ =: 4 4' 

3(PP^^— P^P^3 

From (12), when s = 0, p^ = -— — , which agrees with the above. 

Formula (16), giving the value of the maximum hoop-tension, differs con- 
siderably from that heretofore used, in the deduction of which the longitu- 
dinal stress was considered zero or constant. The existence of a neutral 
surface of longitudinal stress is of great interest in the construction of built- 
up guns. That a longitudinal contraction may accompany a circumferential 
expansion, and must do so under certain circumstances, is a familiar result 
of experience. 

The above is to be found in Vol. VIII, No. 3, of the Proceedings of the U. 
S. Naval Institute ; the notation having been so changed as to make it the 
same as that used in Virgile's formulae. 



SHRINKAGE IN THE TUBES OF A BUILT-UP GUN. 

Up to this point, we have assumed, in all calculations, that each simple 
tube resists at its elastic limit at the moment when the pressure in the pow- 
der chamber is greatest. In order that this shall be so, it is necessary that 
the gun shall be so built that when immersed in any medium, as in air, the 
simple tubes shall be in a state of initial strain; the inner tubes being com- 
pressed, while those near the outer surface of the gun are stretched. It is 
our object now to determine the amount of this strain. 



<; 




In the figure, let the circle of radius A represent a gun-cylinder com 
posed of a number of simple tubes, upon which one or more simple tubes are 
yet to be placed; let O C be the inner radius of a tube to be placed upon the 
first: the quantity A C ■= O A — OC, called the shrinkage, is what we are to 
determine. Now, when the outer tube is expanded by heat, slipped over the 
inner one, and the system is allowed to cool, the surface of contact will have 
some radius O D ; greater than O C. and less than O A. If more tubes are 
similarly put on outside, the radius of contact will become still less, as O E. 
If now, the powder pressure be generated, the radius will increase, as to O F. 

In each of these states, the value of the shrinkage is the same, being, in 
the last, equal to the sum of A F and F C. A F is the amount by which our 
first gun-cylinder would increase in exterior diameter if, at the same instant, 
it were freed from the powder pressure within and the pressure of the tubes 
outside it. F C is the amount the outside tube would contract in diameter 
if the pressures on its two surfaces were similarly made equal. 

These quantities are, respectively, the first and second terms in the second 
member of equation (7). 

P 

By means of Hooke'a Law^ expressed in our notation i = ^, we may, hav- 

ing tabulated values of E, determine i froin p, or vice versa. The value of 
p in the present case requires further investigation, since the stress at any 
point in the mass of the gun is not confined to a plane. 

If a rectangular paraiielopiped, of edges a, 5, c, be subjected to stresses 
parallel to its three edges, its dimensions will be changed. It is clear that, 
within the limits within which the alterations of dimensions are proportion- 
al to the forces causing them, or within which Hooke's Law is applicable, 
the effect of any stress is independent of the existence of prior stress. This 
being admitted, if we denote by i, the elongation produced in the direction 
of a by the force X, and by e, the contraction in the two directions at right 



angles to a caused by the same force; andby ^2, e^ and ^^, e^ the same quan- 
tities caused by forces Yand Z, parallel to band c respectively, we shiill 
have 

^^=^-^-^8 1 

The ratio of e to i is generally admitted to be constant for the same sub- 
stance within the limits of stress considered; its numerical value will be taken 
as i, that being the best determination. We may then write, 






Ac= L\z-k(x-hY)^ I 

Replacing X, — Y. and Z by the quantities T, P. S, in the notation already 
in use, we have for any change in the diameter of a tube subjected to these 
forces, 

E 

We shall in all cases neglect S; the force parallel to the axis of the gun. 

Thelettersi2^,i^^,i^^, .P^.P^.P,. , T^,T^,T^, , T\, T\. 

T' will have the same meaning as hitherto; and we shall call the 

moduli of the plasticity of the various tubes E^. E^,E,^ 

rp ^ rp ^ rp ^ ^ bciug the limits of elasticity of the metals of the various 

tubes, aie known, P^, P^, P^, can be calculated by formula (4), of the 

translation of Virgile; and T\, T\, T ,. may be found from the law that the 
difference of the hoop -tension and pressure at any point is a constant, for ex- 
ample, I _ 

Consider now, the case of a gun consisting of two simple tubes which are 
each stretched to their elastic limit, and thus doing their best work, when 
they have the powder gas, with a known pressure, within, and zero pressure 
without; suppose that the powder pressure is removed, and. at the same in- 
stant, each tube is freed from the other: the outside diameter of the inner 
tube would change per linear unit by the amount 

and the inside diameter of the outer tube would change, per linear unit, 
Therefore, the shrinkage will be, 

Calculation of the Shrinkage at the Various Stages of Construction. 

If the simple tubes were, in practice, each fitted separately, and then put 
together, the method given of calculating the shrinkage would suffice; the 
exterior of each tube, after it has been put in place, is however turned and 
finished before adding another tube, and thus the determination of the shrink- 
age of this tube becomes more complex. .... 

Suppose a tube is in equilibrium under the internal pressure it is intended 



to support, and with its internal surface at the elastic limit, and that this 
condition is, in any manner, modiiied; the pressures and tensions will un- 
dergo variations which we will represent respectively by Po.Pi, P21 P " 

p . ^; ^0' ^11 ^2 i'- t'oj ^17 h^ ^ - 

We shall establish the relations between to and p^, the variations caused 
by the supposed change at the surface of the bore; and f'„ _ i,f,, andp, 
vvhich occur at the surface of contact of any two tubes: we shall thus be able 
to determine the shrinkage which should be given to the second of the two 
tubes. 

The equilibrium of tlie (n + 1) ' tube, in the first state, is given by the 
equations, 

T. — P„ = r — P„4.i; 
the equilibrium of the same tube in the second state, by the equations, 

[{T. + t) + (P,. +p.) Jp;= [(r; 4- O + (^" + 1 +P' + ^0^'h- 1 

[T. + 1) - (P. 4-p.) = [r + t) - (P„ ^ ^ +p_ ^^); 
from these four equations, we derive, 

two relations which might have been anticipated. 
Solving these equations for t' and p . , we have 

P' + p' e\ — p' 

t\ = t, ^^ 3 p,.-^^ -; (1) 

these will give the values of V andp when t, andp, are known. 

""T 1 

The exterior diameter of the {n -\- 1)" tube, in its passage from the first to 
the second state, will undergo a change, per linear unit, of 

the interior diameter of the [n -\-2)' tube will, since the surfaces of the two 
tubes are in eontact, undergo the same change; we have therefor, 

ii/,. \ " '.-1-1/ U_^i\ "+i »+i/ 

whence 

If now, we put n — in (1) and (2), we have 

to =Ao to -i-BoPo, 
Pi = C^ to+D^po: 
Ao, Bo, Ci, Di, being known. Putting n =: in (3), and substituting the 

values of ^0 andp, just found, we derive from (3), 
t, = G,to-hS,po. 
Similarly, making n = 1 in (1), (2), and (3), and using the values of #1, Pi 

ti, and P2, we derive equations of the form 

ti ^^1^0 + -^iPoi 
P2=C2to +D2P0, 
^2 = Gr2 to -^ ^2^0) 



where the cofBcients of to andpo are known. 
We have, therefore, generally, 

/ A t -\-B p , (4) 

'.-1 "-1 «-l 

p.. = ato-hD,.Po. (5) 

t„ = G..to-\-H..Po' (6) 

If now, in building a gun, the first n tubes are in place, and we wish to 
determine the shrinkage to be given the {n + 1) ' tube, we conceive the gun 
to be first in equilibrium with the {n -\- 1) ' tube on, with the powder pressure 
within, and all the tubes at their elastic limit; we then, simultaneously, re- 
move the powder pressure, and free the n tubes from the {n + 1) '. The 
change in the exterior diameter, per linear unit, of the system of n tubes 
will be 



to:-.+i..) 



that of the interior diameter of the [n + 1)" tube 

Calling A the shrinkage per linear unit, we have therefore, 

A. 



1 ((_+tp„)+^(r, + iP.). (7 



"1 

Now, to represent analytically the passage of the system of {n -\- 1) tubes, 
from the first state, that of resisting together at their elastic limit the powder 
pressure, to the second, that without the powder pressure and of separation 
at the n"' surface of contact, we havepo = — -^oi and p, = — P„m (4), (5), 
and (7). Thus (7) becomes, 

A„:=^ A' -iP„)+-^f r„-f iP„V (8) 



to compute f'_^, we find to^ by substitution of the known values of p^ and p^ 
in (5), and this value, v/ith thatof Poi ^^ (4), gives f_i; which finally, in (8)' 
gives A. 

In the case in which the tubes are of the same metal, we have EZ^ = E,. 
= £^-, whence, from (8), 



A„ 



iOL + ^") (^) 



also we have 



whence 






(10) 



2po Ro-pAR -\-Ro^ 



-K„ — -Ko 
if, in this equation, we make Pq-=z — P^ and p„ = — P„, we find 

Tangential Compression at the Surface of the Bore, 

When a built-up gun is not subjected to powder pressure within, the inside 
surface is compressed; and, as the amount of this compression might become 
so great as to injure the metal at this point, it is important to ascertain iis 
value in any given case. In the case of sudden alterations of pressure, vibra- 
tion would be set up, and the danger of injury would be increased as shown 



in the determination of the effect of a sudden force. 

If the gun in question consists of n tubes, we have, from (5), by putting 
p.. =: 0. and Po = — Po 

whence 

t, = '^. (12) 

In this equation, to is the negative change of hoop-tension which the sur- 
face of the bore undergoes in passing from the extreme tension to the extreme 
compression. T^, being the first of these limits, the second will be represen- 
ted by the algebraic sum Tq -p ^o- This is a negative quantity whose abso- 
lute value is the compression per linear unit. 

If all the tubes have the same elasticity, we have, from equations (10), 

«„=-P„^i±^ (13) 

Rn JRo 

Examples. 

Cast-iron Gun icith Hoops of Steel. 

Suppose the gun has one caliber's thickness of cast-iron, and one-half cali- 
ber's thickness of steel outside; the elastic limits of the cast-iron and steel 
are 8 kilos and 25 kilos per square millimeter; the moduli of the two metals 
are 10.000 and 20,000 kilos respectively. As shown on page 20 of the trans- 
lation of Virgile, Po« is 19 kilos; and Pj is 7 kilos per square millimeter: 
hence 

r; = ro-PoH-Pi=-4. 

The change, per linear unit, of the exterior diameter of the cast-iron tube 
will therefore be 

~ t"tji — — .000 166, 

10.000 

that of the interior diameter of the steel hoop 

25-h ^ 



= -h .00136- 



20.000 

The sum of these two quantities, or .001533, will be the shrinkage required 
per linear unit. 

To find the compression of the inner surface of the bore. The cofficient of 
fo and_Poi i^ (1) ^^^ ("^l ^^e 









2 PI 


— 9 




















r\-rI 


4 




















2Rl 


— y 














We have 


therefore, from 


(1) and (2) 




















t=lto- 


tPo. 




















Pi=-tt 


3 + e.Po- 














Since, 


in 


the 


case considered, — "— — = 




we have, 


from 


(3) 












^.=2/-;-+ 


ipl^ 














but 






(2/o = V^o-fPo 




















hp:=- 


-^l^^o+^ 


-%~ 


Po 










hence 






t,=^¥' 


-to-H'Pc 


• 












Putting, 


in 


(2), 


vi=l, Pi = 


3, p2 = 4; as indicated by the data of 


the 


prob 


lem. we 


find 
























P2=— 3^2 


^i+!fPi 














lint 






j-sV^: 


= -fHUo 


+ 


mhPo 










UUt 






I IfjPl 


==-^hto 


+ 


HmPo 











hence 

^^ W. 0: H 9 / I 1 (i 9 ■ .J I- ^. 

p.j, !. -v 3 - t'O n^ i' A- .; J 2-^0 • 

In this equation, we make p2 = 0, and 2>i, '— — 19; whence we find 

#0— — 20.0. 
The tension of the inner surface of the bore when the internal and external 
pressures are the same is therefore 

f + 'i'o = — 20 -f 8 — — 13. 
This is the force of compression in kilos per square millimeter at the sur- 
face of the bore. It is well within the strength of the metal.. 

This example can be otherwise worked out by finding the value of ^o from 

(12). and then using (4) with the value of f „ just found to find t ^, we finally 
substitute the last in (7) to find A, In the first operation it is to be noticed 
that the value of n for which p= is n =: 2. (12) therefore becomes 

The coefficients are easily found, as indicated in deriving (4), (5), and (6); 
and F(. is the internal pressure. 

For a second example we will take a gun made of two steel tubes and a 
row of steel hoops: the walls being one caliber thick, and the successive radii 
increasing in geometrical proportion. The modulus of elasticity is the same 
for all; namely 20,000 kilos per square millimeter, and the tubes are to be set 
to pull at 20 kilos and the hoops ai 25 kilos per square millimeter at the mo- 
ment of maximum tension. 

Making i?n - 1; we find J?, = 1 .44. F, = 2.07. and R. — 3. We shall also 
find P,_ = 8.72. P, :=--: 18.73. and P., ^ 32.24. 

For the shrinkage: 

In formulae (11) and (9i. making successively n -.=- 1 and ii =2. we find: 
t'^ = — 6.49 and A^ = .000673. 

t'^s=z — o.Q2 and -\ , = . 000969 ; 
the two tubes should therefore be adjusted with .00068 shrinkage per unit of 
length of diameter, and the hoops with .00097. If the gun were of 6 inch 
caliber, we find i2„ = 3. _R, — 4.3. Zy. — 6.2 jR,--9: for the difierence of 
diameter at the surface of contact of the two tubes. .003 inch and at the other 
surface .006 inch 

Compression of bore: 

Making n=zS in (13). we find 
fo= — 40.3. 
whence T,, -\~t^ = — 20.d. 

Thus, in round numbers, we vary the tension of the metal at this point 
from -f-20 to — 20 kilos per square millimeter, or from -i- 28.446 to — 28.446 
lbs. per square inch (1 kilo per square millimeter = 1422.3 lbs. per square 
inch). The most convenient way of reducing this dangerous excess >vould 
be to decrease the initial strain of the steel hooj). 



'/: 



ON THE STRENGTH OF A THICK HOLLOW SPHERE. 

Let the figure on page 6 of Virgile represent a diametral section of a hol- 
low sphere, the notation being the same as given there. The pressure to be 
resisted at the section is 

^{PR^—P'R'^); 
and if the section of the metal be conceived to be divided into an indefinite 
number of concentric rings, the breadth of one of these rings being dp, it 
appears that the total resistance of the section will be 

2 TT I tp dp\ 
J R 
and hence the equation to be fulfilled will be 
*R' 

tpdpz=PR^-P' R'^ (1> 

R 

Let t = - — ' in (1) ; and we have, after integration 

2f{R') — 2f{R) = PR^—P'R'^', 
whence, since R' and R are independent of each other, in general 

2f{p)=-pp\ 

and 2f{p)=—2pp—p^-^-=:2tp (2). 

As in the case of a thick hollow cylinder, the solution is effected by two par- 
tial solutions. From symmetry it appears that the axes of stress at any par- 
ticle must be. one in the direction of a radius, with the pressure p along it, 
and the other two in any two directions perpendicular to the first and to 
each other, with equal tensions t along them. 
For the first case, being that of a fiuid tension^ we put — p = m: whence 

dp 
we find, from (2), - - =r O, and t = — p. Substituting these values in equa- 
tion (1), reduces both members of that equation to 

m{R'^-—R^). 
For the second case, being that of a pair of circumferential tensions, each 
equal to half of the radial pressure, we put_p = 2 ^ in (2), whence 

.__£_ dp 

*- 6- >""• ^^^ 

T 4. dp 6a ... 

Let — — = —, (4) 

dp p' 

then substituting in (3), we have 

Substituting the values of t andp from this equation in (1), we find, after 
integrating. 

This equation gives c = 4; 

From (5) then, we see that the second case gives . 

t — — — ^ 
P^ ~ 2' 
Combining the two solutions we have, 

^=^-f-m| (6). 

2a \ 

whence {t-\-p) p^ = ?>a\ (7) 



2t 



9) p3_3al 
p=:3m ) 



The constants a and w maybe determined from (4) and (6). Their values 
are 

_ {P — P')R' E'^ 

PR^ — P'R'^ 



m 



R'^—R^ 
From the first of (6), we derive 

■R' \ // 2(P4-T) 



=V(^ 



R " ^ \2TH-3P'— P/ ^^^ 

Since P', in shells, is very small when compared to P, this equation shows 
us that, if P is equal to, or greater than 2 jT, no thickness of metal will be 
sufficient to withstand the pressure. We may also solve this equation for 
P, thus obtaining a formula which would enable us to proportion blowing 
charges in shells. 
The above result is taken from Professor Rankine's Applied Mechanics. 



GUN DESIGNING. 

A gun. like all other machines, must be designed to fulfil certain definite 
conditions. Its projectile may be required to pierce a given thickness of 
armor at a given distance; or weight of piece maj' be the limit, and it may be 
wished to throw the most powerful shell or shrapnel to a given distance with 
a given elevation, consistent with that limit. To work out problems of this 
nature, it is all-important to possess an accurate knowledge of the action of 
the charge inside the bore. By means of the Noble chronoscope and the 
crusher gauge, this knowledge is obtained, and w^e shall now explain how^ 
the indications of these instruments are employed to assist in determining 
the proportions of ordnance. 

Gunpowder is not properly so much an explosive as a substance burning 
and giving off gas with great rapidity. It offers in this respect a marked 
contrast to gun-cotton, dynamite, and other true explosives. If one of these 
agents be detonated, the detonation is immediately carried through the mass, 
whatever its size, and the whole at once turns into gas. Gunpowder, on the 
other hand, as far as is known at present, cannot be detonated, but simply 
evolves its gas by burning in layers from the outside to the inside. Thus a 
large grain will take longer to burn up and become entirely converted into 
gas than a small one w411; hence the effect of enlarging the grains is to render 
the action of the charge less violent, the composition of the powder being the 
same in all gun charges. The projectile is driven out of the bore by the pres- 
sure of the gas on its base — that is. on an area that varies with the square of 
the calibre. The weight of projectiles of similar form varies with the cube 
of the calibre. Hence the larger the gun the heavier will be the column of 
metal or projectile driven by each square inch of its base; and the great'er 
must be either the pressure applieJ, or the time of its application, if a given 
velocity is to be attained. The great object of the gunmaker is to obtain the 
highest possible ratio of muzzle velocity to breech pressure. His ideal would 
be a charge so arranged that a pressure equal to the amount the gun is con- 
structed to bear should be uniformly maintained till the shot has left the 
muzzle. Science is still a long way from this, but has done a good deal towards 
it in the last few year^. A charge of gunpowder, composed of service ingre- 
dients, in service proportions, exploded in a closed vessel at a density of 1.00 
(equal to that of water), sets up a pressure of 43 tons per square inch: at a 
density of 0.75. of 23.2 tons; at 0.50 of 11.8 tons. Supposing a gun cartridge 
to be rammed home to the density of water, and entirely converted into gas 
before the projectile began to move, the pressure in the bore would rise to 
43 tons per square inch at the breech, and fall towards the muzzle, as the 
travel of the shot afforded increasing room for expansion behind it. The 
column of metal to be moved, even in the heaviest projectiles yet known, is 
only a few pounds to the square inch of base, while the.maximum pressure 
of the powder gas is measured in tons; it is clear therefore that the shot must 
get under way at some period antecedent to the setting up of the maximum 
pressure. In a breech-loader, where the projectile has to be forced through 
a bore slightly less than its greatest diameter, it will be detained longer than 
in a muzzle-loader, where it moves freely away, but the difference is insig- 
nificant as regards the present argument. The result of the shot's early mo- 
tion is that space is at once given for expansion, and the normal 43 tons is 
never reached. Before these matters were fully understood, badly propor- 
tioned charges of violent powder were found sometimes to set up what are 
known to artillerists as "wave pressures," which were dynamical in charac- 
ter, being caused by rushes of gas from one end of the charge to the other, 
so that the gauges indicated far higher pressures at the ends of the powder 
chamber than in the centre. This has now been overcome, and a great in- 
crease of both power and safety has been obtained. Several important im- 
provements have been made of late years; the principal ones are three in 
number: — (1) a great stride was made in the manufacture of powder when 
pebbles, prisms, and li inch cubes were introduced; (2) the discovery of the 
beneficial effect of ''chambering," that is of boring out the powder chamber 
to a greater diameter than the rest of the bore; (3) the method of ''air spac- 
ing" the cartridge, so that a certain weight of powder should have a certain 
definite space allotted to it, irrespective of the actual volume of the powder 
grains. Thus in the 80 ton gun powder cubes of H inch edge are used, having 
an absolute density a little over 1.75, or about 15.7 cubic inches to the pound. 



ff these grains were rammed tightly home in a silk-siioiub bag. the space occu- 
pied behind the shot would be 24.6 cubic inches per pound; as actually used, 
an air-space over and within the cartridge is left, so that the space behind 
the shot amounts to 34 cubic inches per pound. This density would set up a 
pressure in a closed vessel of 26.6 tons per square inch, but the relief afforded 
by the shot's motion reduces it to about 19 tons per square inch. The effect 
of chambering out the end of the bore where the powder lies is practically 
|:o permit a small gun to consume effectively the charge of a larger one. The 
«*artridge is shortened, and the mechanical conditions of burning are greatly 
sraproved, so that with large charges, higher velocity with lower pressure is 
obtained from a chambered than from an unchambered gun. The above in- 
formation is derived from the indications of the crusher gauge, which regi.^- 
ters the pressure of the gas at various parts of the bore. The chronoscope 
measures the rate at which the projectile acquires velocity during its travel 
from the breech to the muzzle. Knowing the increment of velocity at any 
point, we can calculate the amount of pressure required to produce this in- 
crement, and thus confirmation is obtained of the accuracy of the records 
obtained by the crusher gauge. The following table gives the increase ob- 
tained at successive stages in the developm.ent of the power of the 80 too gun. 
which was first under-bored for experiment, and gradually brought to it:^ 
present dimensions: — 



i o 



b 


b 


b 


1— I 
or 

o 1 

! 


or 

b 


C?l 

b 


! 


i" 


1 


j 

00 pO C5 p 0» 

b b b b b 


? 1 ? 

O 1 Ol 

1 


s 


g- 5 i 


li 


5^ 


CC CO to »o to 

c;t .-i. o to to 
O O O j o o 

1 1 


3D 




CO 

*>- 


^ pi p p ^ 

b b b b b 


b 






-5 


. .1^ 

^ -^ >^ 

O 1 O 1 Ci 

CO j CO OS 


Ci 


X 1 to 


3= 

OB 


£.2. i 

?a 1 


i ^ 

! 


CO 1 rfi- 

K 

! 


to 
to 


to 

to 


to 

CO 
CO 


to 

«o 


1 




OS 
i.O 


en 07 
00 O 
oo zn 


C?t Oi Mi' or 
OT h-^ to O 
00 H-i CO to 


? 1 Is 


29,745 
31,527 


i 


! 


i 




t if \ 

an 7? CD 1 


i s 


P 

b 


cc 

b 


o 

CO 

CO 
b 


b 


b 


CO 
b 


■ 


111 


i 


or 


1 


or 




CO CO 


jag-? 

72 Hj 




i 


^ 

^ 


P 


5^ 

GO 


^ 


p 


P 


P' 


0<j. ^ o 

P ® o °g 



I— ( 
O 

i-H 

H 

■-73 



oo 
o 

O 



It will be obserA^ed that each improvement has tended to facility of con- 
Kumption of increased charges, so that while the pressures are diminished, 
the penetrating power of the projectile is augmented, a heavier and more 
destructive shell being driven through thicker armor. 

The manner in which the various principles, of which an explanation has 
been above attemjjted, are practically applied is laborious and complicated: 
the conditions are often conflicting, and the ultimate dimensions of a piece 
of ordnance is commonly a compromise. A couple of simple examples will 
illustrate the modus operandi. Suppose that it is required to design a gun 
which shall not exceed a given length, but shall throw a projectile capable 
of piercing a given thickness of iron at a given range. There are several 
formulas of a more or less empirical nature for calculating the perforating 
power of a projectile moving with a known velocity. Penetration is by some 
regarded as a punching action, by some as a wedging action; probably it is a 
compound of the two. Recent experiments carried on with the very high 
velocity of about 2000 f. s. ha ? thrown some doubt on the soundness of any 
of the formulai, That generally used in England is as follows: — 

Let W = weight of projectile in tons; 

r =r radius of projectile in inches; 
V = velocity of projectile in feet per second; 

.- E= — = energy of projectile in foot tons; 

- t = thickness of plate perforated in inches; 

then -— z=2.52.Y 1:«, 

2 r* 

This formula tells the gun-designer what energj- is necessary to overcome 
the resistance of the jjlate. Guided by experience he assumes for the moment 
a striking velocity; the other component of the energy, the weight of the pro- 
jectile, is then directly arrived at. The proportions of armor- piercing shell 
are the same for all guns so that the weight guides the dimensions, and the 
calibre of the gun follows. Should this appear to be in no way unsuitable to 
the length already laid down in the conditions, the gun-designer calculates 
the loss of velocity in the given range and from the striking velocity deduces 
the muzzle velocity and the muzzle energy. The excellent labors [Eesearches 
OH Explofiivef<) of Captain A. Noble. F. R. S., of Elswick. and Professor Abel. 
F. R. S.. have shown how to calculate the amount of work done by a pound 
of jDowder for every volume of expansion its gas undergoes; theresults of 
many c ireful experiments and much intricate calculation are embodied in 
the table on page 165, Text Book of Gunnery, Captain G. Mackinlay, R. A., 
which affords the means of determining the tot.ii work performed by any 
charge in any gun. A certain portion of this work is expended in heating 
the gun and projectile in giving rotation and so forth: the remainder appears 
as the energy of translation of the shot on leaving the muzzle. Large guns 
realize a greater proportion of the total work than small ones: the gunmaker 
knows very approximately by experience what percentage may be expected 
from certain classes of ordnance with certain descriptions of powder. 

Roughly the factor of effect may be put within the following limits: — boat 
guns, 60 to 65 per cent.: medium guns (such as the 60 pdr. and 80 pdr. B. L. 
R's., and the 8 in. M. L. R.), 70 to 80 per cent.; heavy guns. 85 to 95 per cent. 
The method of calculation will be best understood from an example. Sup- 
pose a charge of 425 fos. of P.y powder is to be tired from the 80 ton gun cham- 
bered to 18 inches diameter: the projectile weighs 1700 tt)s.. and the space 
behind it is 14, 450 cubic inches. The whole content of the bore is 60, 400 
cubic inches, and the volume of the charge is (425. X 27.7) 11, 773 cubic inches, 
the number of expansions therefore is 5.13; the table (page 165, Mackinlay.) 
shows that powder gas expanding to this extent from a density equal to that 
of w^ater can perform work amounting to 92.4 foot tons per lb. Since, how- 
ever, the charge burns up in and has to fill (425 X 84) 14, 450 cubic inches 
before doing work, the energy due to this extent of expansion (1.227) is lost, 
and 17.7 foot tons per ft. must be deducted, leaving (92.4 — 17.7) 74.7 foot 
tons per ft. as the total work the charge is capable of performing under these 
conditions. It is known from the preliminary tests of the powder that in the 



so ton gun between 92 and 95 per cent, of the total work will be realized. 
Hence the energy of the projectile will lie between 29, 210 and 30, 155 foot 
tons, and its muzzle velocity betueen 1581 and 1600 f. s. A reference to the 
table in this article will show that the result actually arrived at lies nearly 
midway between these limits. In this manner the charge required to impart 
the necessary energy to a shot of given weight in a given length of bore, and 
conversely, the length of bore which will contain the requisite number of 
expansions of a given charge, are easily found; hence the charge required to 
produce the necessarj^ energy is readily found; the air-space and the dimen- 
sions of the powder chamber follow, and the inside of tJie gun is settled. The 
gun-designer now has to put walls round his bore. 

Guided by the knowledge previously mentioned as derived from the crusher 
gauge and the chroaoscope, he lays do wa the pressure-; at each point of the 
interior, and calculates the amount and strength of the metal to be used, 
according to the sj)ecial system of construction employed, and thus the ex- 
terior of the gun is settled. 

To give another instance: — let it be required to make a gun of moderate 
power, not to exceed 17000 1t;s. in weight: recoil (of piece only) must not 
exceed 15 feet per second; the requirements are further, that the gun must 
have a range of 1300 yards with not more than 2'^- elevation, and 4000 yards 
with not more than 8- elevation. A gun weighing 17000 tt^s. and recoiling 
at 15 f. s. velocity gives 255.000 units of momentum (in pounds and foot- 
seconds): the iuitial momentum of the gun ic lecoiling is practically equal 
to the momentum of the shot on leaving the muzzle. Hence the shell must 
have 255.000 units of momentum, which may be composed of high velocity 
and light weight, or low velocity and heavy weight. For general service, 
accuracy and penetration are essential qualities: a shell varying in length 
from tvv-o and a half to three times its diameter v.'iii be suitable for ordinary 
purposes. We have now the following problem in ballistics, viz.: To find 
the calibre of a sljell of the proper length, of sucli a weight that, with the 
muzzle velocity required to give a range of 1300 yards at 2- elevation, the 
muzzle momentum shall be 255.000 units. This problem is readily solved by 
the methods indicated in chapter on trajectories and it is found that a shell 
8 in. in diameter, weighing 180 t^ s.. and halving a muzzle velocity of 1400 f s.. 
will about fill the requirements. Proceeding in the same manner we find 
that a projectile of the same dimensions and having the same muzzle velocity 
will satisfy the second requirement. The calibre being thus settled, the pro- 
portions of the piece remain to be worked out. There is little difficulty in 
obtaining a velocity of 1400 f. s.. with a moderate charge and moderate pres- 
sures, so the length of the bore and the disposition of the metal can be adjust- 
ed to suit the strain of discharge. In the gun under consideration, the bore 
should be made as long as possible, and the weight of metal thrown as far 
forward as possible consistently with preserving due strength at the breech. 
The reasons for this are twofold: — first, the longer the bore the less is the 
breech pressure required to produce a given, muzzle velocit3^ and the less is 
the maximum strain thrown upon the rotating agent; secondly, the more for- 
ward the general disposition of the metal, the farther from the breech end 
will be the cetre of gravity, and consequently the trunnions — a position which 
favors steady shooting and absence of jump. 



[From, Gunmaking and Gunnery, Col. E. Maitland, R. A. Encyclopedia 
BritannicaJ. 



f)JF 



rrM 



■3 
IP 



>jx^ 



!)■'£> 



~^1)> 



1I> 










































,p:>II> ^3>,,^ -^.C^ -^A^ -^^=2^^' 






<iv>^^^ " ^?>t> -y^ :m> y^* ,i>s>:>'i? 



>1> 









' "3i> o^^' "m) ■ ^ 



>^0 -yp '^J>' 



3^ 






.^ .^ 2>3 





















3 :xi>^ >:!>.«=■ 






^^la^'- 


-K^^ 


>>:^>^ 


'jS> L^.-^